Automorphic Forms And Shimura Varieties Of Pgsp(2).
Title:
Automorphic Forms And Shimura Varieties Of Pgsp(2).
Author:
Flicker, Yuval Z.
ISBN:
9789812703323
Personal Author:
Physical Description:
1 online resource (338 pages)
Contents:
CONTENTS -- PREFACE -- ACKNOWLEDGMENT -- PART 1. LIFTING AUTOMORPHIC FORMS OF PGSp(2) TO PGL(4) -- I. PRELIMINARIES -- 1. Introduction -- 2. Statement of Results -- 2a. Homomorphisms of Dual Groups -- 2b. Unramified Liftings -- 2c. The Lifting from SO(4) to PGL(4) -- 2d. Special Cases of the Lifting from SO(4) -- 2e. The Lifting from PGSp(2) to PGL(4) -- 2f. Elliptic Representations -- 2g. Automorphic Representations -- 2h. Unstable Spectrum -- 2i. Generic Representations -- 2j. Orientation -- 3. Conjectural Compatibility -- 4. Conjectural Rigidity -- II. BASIC FACTS -- 1. Norm Maps -- 2. Induced Representations -- 3. Satake Isomorphism -- 4. Induced Representations of PGSp(2,F) -- 5 . Twisted Conjugacy Classes -- III. TRACE FORMULAE -- 1. Twisted Trace Formula: Geometric Side -- 2. Twisted Trace Formula: Analytic Side -- 3. Trace Formula of H: Spectral Side -- 4. Trace Formula Identity -- IV. LIFTING FROM SO(4) TO PGL(4) -- 1. From SO(4) to PGL(4) -- 2. Symmetric Square -- 3. Induced Case -- 4. Cuspidal Case -- V. LIFTING FROM PGSp(2) TO PGL(4) -- 1. Characters on the Symplectic Group -- 2. Reducibility -- 3. Transfer of Distributions -- 4. Orthogonality Relations -- 5. Character Relations -- 6. Fine Character Relations -- 7. Generic Representations of PGSp(2) -- 8. Local Lifting from PGSp(2) -- 9. Local Packets -- 10. Global Packets -- 11. Representations of PGSp(2, R) -- 11a. Representations of SL(2, R) -- 11b. Cohomological Representations -- 11c. Nontempered Representations -- 11d. The Nontempered: -- 11e. The Nontempered: -- VI. FUNDAMENTAL LEMMA -- 1. Case of SL(2) -- 2. Case of GSp(2) -- 2a. Preliminaries -- 2b. L(F) is a Product -- 2c. L(F) is a Quartic Extension -- PART 2. ZETA FUNCTIONS OF SHIMURA VARIETIES OF PGSp(2) -- I. PRELIMINARIES -- 1. Introduction -- 2. Statement of Results -- 3. The Zeta Function -- 4. The Shimura Variety.
5. Decomposition of Cohomology -- 6. Galois Representations -- II. AUTOMORPHIC REPRESENTATIONS -- 1. Stabilization and the Test Function -- 2. Automorphic Representations of PGSp(2) -- 3. Local Packets -- 4. Multiplicities -- 5. Spectral Side of the Stable Trace Formula -- 6. Proper Endoscopic Group -- III. LOCAL TERMS -- 1. Representations of the Dual Group -- 2. Local Terms at p -- 3. The Eigenvalues at p -- 4. Terms at p for the Endoscopic Group -- IV. REAL REPRESENTATIONS -- 1. Representations of SL(2, R) -- 2. Cohomological Representations -- 3. Nontempered Representations -- 4. The Cohomological L(vsgn, v-1/2 2k) -- 5. The Cohomological L(v1/2 2k+1, v-1/2) -- 6. Finite Dimensional Representations -- 7. Local Terms at -- V. GALOIS REPRESENTATIONS -- 1. Tempered Case -- 2. Nontempered Case -- PART 3. BACKGROUND -- I. ON AUTOMORPHIC FORMS -- 1. Class Field Theory -- 2. Reductive Groups -- 3. Functoriality -- 4. Unramified Case -- 5. Automorphic Representations -- 6. Residual Case -- 7. Endoscopy -- 8. Basechange -- II. ON ARTIN'S CONJECTURE -- REFERENCES -- INDEX -- AUTOMORPHIC FORMS AND SHIMURA VARIETIES OF PGSp(2).
Abstract:
The area of automorphic representations is a natural continuation of studies in the 19th and 20th centuries on number theory and modular forms. A guiding principle is a reciprocity law relating infinite dimensional automorphic representations with finite dimensional Galois representations. Simple relations on the Galois side reflect deep relations on the automorphic side, called âliftings.â This in-depth book concentrates on an initial example of the lifting, from a rank 2 symplectic group PGSp(2) to PGL(4), reflecting the natural embedding of Sp(2,?) in SL(4, ?). It develops the technique of comparing twisted and stabilized trace formulae. It gives a detailed classification of the automorphic and admissible representation of the rank two symplectic PGSp(2) by means of a definition of packets and quasi-packets, using character relations and trace formulae identities. It also shows multiplicity one and rigidity theorems for the discrete spectrum. Applications include the study of the decomposition of the cohomology of an associated Shimura variety, thereby linking Galois representations to geometric automorphic representations. To put these results in a general context, the book concludes with a technical introduction to Langlandsâ program in the area of automorphic representations. It includes a proof of known cases of Artinâs conjecture.
Local Note:
Electronic reproduction. Ann Arbor, Michigan : ProQuest Ebook Central, 2017. Available via World Wide Web. Access may be limited to ProQuest Ebook Central affiliated libraries.
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